Method of Detecting System Function by Measuring Frequency Response

ABSTRACT

Methods of rapidly measuring the impedance spectrum of an energy storage device in-situ over a limited number of logarithmically distributed frequencies are described. An energy storage device is excited with a known input signal, and the response is measured to ascertain the impedance spectrum. The excitation signal is a limited time duration sum-of-sines consisting of a select number of frequencies. In one embodiment, magnitude and phase of each of frequency of interest within the sum-of-sines is identified when the selected frequencies and sample rate are logarithmic integer steps greater than two. This technique requires a measurement with a duration of one period of the lowest frequency. In another embodiment, where the selected frequencies are distributed in octave steps, the impedance spectrum can be determined using a captured time record that is reduced to a half-period of the lowest frequency.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of co-pending U.S. patentapplication Ser. No. 12/217,013, filed Jun. 30, 2008, which is acontinuation-in-part of U.S. patent application Ser. No. 11/825,629,filed Jul. 5, 2007, now U.S. Pat. No. 7,395,163 B1, issued Jul. 1, 2008,which is a continuation of U.S. patent application Ser. No. 11/313,546,filed Dec. 20, 2005, now abandoned, which claims the benefits of U.S.Provisional Patent Application Nos. 60/637,969, filed Dec. 20, 2004 and60/724,631, filed Oct. 7, 2005. This application further claims thebenefits of U.S. Provisional Patent Application No. 61/186,358; filedJun. 11, 2009. The disclosures of each of these applications are herebyincorporated by reference in their entirety, including all figures,tables and drawings.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with Government support under Grant No.DE-AC07-05ID14517 awarded by the United States Department of Energy. Thegovernment has certain rights in the invention.

REFERENCE TO SEQUENCE LISTING, A TABLE, OR A COMPUTER PROGRAM LISTINGCOMPACT DISC APPENDIX

Not Applicable.

BACKGROUND OF THE INVENTION

Electrochemical impedance measurement systems generally use Bodeanalysis as a well established and proven technique to characterize theimpedance of an energy storage device (ESD). The ESD under evaluation isexcited with an input signal at a given frequency, and the response ismeasured. This process is sequentially repeated over a range offrequencies until the impedance spectrum is obtained. This method iseffective in assessing ESD degradation over time and usage, but itrequires expensive laboratory equipment, and it can be time consuminggiven the serial measurement approach.

An alternative approach using bandwidth limited noise as an excitationsignal can also be used to obtain the impedance more quickly. The systemresponse to the noise is processed with correlations and Fast FourierTransform (FFT) algorithms. This technique was developed at the IdahoNational Laboratory (U.S. Pat. No. 7,675,293 B2) and successfullyapplied to various battery technologies (Christophersen, 2008). However,this approach requires the average of multiple measurements toadequately determine the impedance response over the desired frequencyrange, which also makes it more of a serial approach.

Rapid, in-situ acquisition of ESD impedance data over a desiredfrequency range can be implemented with Compensated SynchronousDetection (CSD) and Fast Summation Transformation (FST). Unlike typicalAC impedance measurements and noise analysis methods, these techniquesare parallel approaches that require only a single time record tocapture the ESD response. As a result, both CSD and FST are well-suitedfor onboard applications that require impedance measurements as part ofan overall smart monitoring system used for control and diagnostics.

Compensated Synchronous Detection (U.S. Pat. No. 7,395,163 B1) is atechnique that inputs an excitation signal consisting of a select numberof logarithmically distributed frequencies in a sum-of-sines (SOS)configuration. The duration of the SOS excitation signal depends on thefrequency step factor and the desired resolution of the impedancespectrum. Typical CSD measurements require a minimum of three periods ofthe lowest frequency (Morrison, 2006). A time record of the ESD responseto the SOS excitation signal is also captured at an appropriate samplerate. The resulting data are used to calculate impedance usingsynchronous detection (Lathi, 1998) to estimate the “In Phase” and the“Quadrature” components at each frequency. Because of crosstalk error, aCompensation Time Record (CTR) is created by reassembling all of thedetected responses except for the frequency of interest. This suppressesall of the other frequency components and allows the frequency ofinterest to be detected with greatly reduced corruption from crosstalk(Morrison, 2006). The CTR is then subtracted from the originallycaptured response signal, and the result is synchronously detectedagain. This process is repeated at each frequency of interest to achievean impedance spectrum with minimal error. The “In Phase” and“Quadrature” components can be easily converted to magnitude and phaseangle with simple trigonometric relations.

Fast Summation Transformation (Morrison, 2009) is a variation of CSD. Itis based on a SOS input signal using octave harmonics to cover thedesired frequency range (Morrison, 2008). Since crosstalk error iseliminated with octave harmonics, only one period of the lowestfrequency is required to complete the measurement and obtain theimpedance spectrum. A time record of the ESD response to the SOSexcitation signal is also captured at an appropriate sample rate. TheFST detection process obtains the “In Phase” and “Quadrature” componentsby first rectifying the response signal relative to the sine and cosineat each frequency of interest, adding up all the data points in therectified signal, normalizing to the number of periods of the givenfrequency, and then storing that result. Results from these rectifiedresponses are placed in a two-element vector (sine and cosine), andmultiplied by a conversion matrix (Morrison, 2008) to yield the “InPhase” and the “Quadrature” components at the desired frequency. Thisprocess is repeated for each of the octave frequencies in the SOS. The“In Phase” and “Quadrature” components can be easily converted tomagnitude and phase angle with simple trigonometric relations.

BRIEF SUMMARY OF THE INVENTION

The invention involves using a parallel approach to analyze theimpedance or other related system functions. The analysis methodologiesdescribed herein are variations of the Fast Summation Transformation(FST) technique. A number of logarithmically distributed frequencies areselected and assembled into an Excitation Time Record (ETR) thatconsists of a sum-of-sines (SOS) signal. Next, the ETR is conditioned tobe compatible with the energy storage device (ESD) under evaluation. TheESD is then excited with the ETR, and the Response Time Record (RTR) iscaptured. The RTR is synchronized to the ETR, and processed by a seriesof equations to obtain frequency response.

In one preferred embodiment, Generalized Fast Summation Transformation(GFST), the test frequencies included in the SOS signal are increased byan integer greater than two (i.e., N>2, whereas FST is based on N=2). Astart frequency is chosen and multiplied by the integer step factor toget the next test frequency, and this process is repeated until thedesired frequency range is covered. The resulting test frequencies arethen assembled into an ETR as a SOS signal that has a duration of oneperiod of the lowest frequency. The ETR is used to excite the energystorage device, and the response time record is simultaneously capturedat an appropriate sample rate (e.g., N² times the highest frequency inthe SOS). The RTR is then processed with the GFST equations describedbelow to estimate the magnitude and phase for each test frequency.

In another preferred embodiment, Reduced Time Fast SummationTransformation (RTFST), the test frequencies are established based onoctave harmonics (i.e., N=2). The assembled ETR, however, only requiresa duration of a half-period of the lowest frequency (whereas FSTrequired a full period of the lowest frequency). As with FST and GFST,this method also has an ETR that is based on a SOS signal. However, theRTFST signal requires that the second-lowest frequency be eliminatedfrom the SOS composite. For example, if the lowest frequency is 1 Hz,then the SOS signal would include 1 Hz, 4 Hz, 8 Hz, 16 Hz, etc. whileomitting the signal at 2 Hz. The ETR is used to excite the energystorage device, and the response time record is simultaneously capturedat an appropriate sample rate (e.g., four times the highest frequency inthe SOS). To calculate the impedance spectrum, the RTR is first copied,inverted and then concatenated with itself. This new RTR is processed bythe RTFST equations described below to estimate the magnitude and phasefor each test frequency except the second-lowest frequency. If themissing frequency component is desired, a new ETR is generated with boththe lowest and third-lowest frequencies omitted. Using the aboveexample, the new ETR would incorporate frequencies at 2 Hz, 8 Hz, 16 Hz,etc. while rejecting the signals at 1 Hz and 4 Hz. In this case, the ETRwill have a duration of a quarter-period of the lowest frequency.Consequently, an impedance spectrum obtained through RTFST can beachieved in half the time as FST if the second-lowest frequency is notrequired. If it is required, the RTFST can still acquire the impedancespectrum with a reduced time period (0.75 times the speed of FST).

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

FIG. 1 is a flow chart showing a preferred embodiment of the method ofthe subject invention.

FIG. 2 is a flow chart showing another preferred embodiment of themethod of the subject invention.

DETAILED DESCRIPTION OF THE INVENTION

The method of the subject invention allows for rapid measurement of theimpedance for energy storage devices. It has been shown that the shiftsin the impedance spectra as a function of time and use stronglycorrelates to the health of, for example, battery technologies(Christophersen, 2002). Therefore, the subject method providesinformation about energy storage devices that is critical for onboarddiagnostics and state-of-health estimation. The subject method measuresthe frequency response of an ESD (e.g., a battery) by exciting it with aSOS signal consisting of a number of select frequencies, and thencapturing a response time record. The data are processed to obtain theimpedance at each frequency in interest.

The desired frequencies are assembled as an excitation time record thatcan consist of a sum-of-sines signal with a length of, at most, oneperiod of the lowest frequency. The response signal must be measured ata time step that is compatible with Shannon's sampling constraints forthe highest frequency component. The individual waveforms could be sinewaves of equal amplitude but with alternating phase shifts of 180degrees between each frequency components. Alternating phase shift helpto minimize any start-up transients. The Root Mean Square (RMS) and therogue wave peak (i.e., the sum of the absolute values of all componentpeaks) of the assembled time record must be compatible with the ESDbeing excited and the Data Acquisition System (DAS) that will capturethe response.

The excitation time record is first signal conditioned to be compatiblewith the ESD under test. As part of the signal conditioning,anti-aliasing filters ensure that only the intended frequencies arepassed, while all other frequencies generated by the digital to analogconversion process are suppressed. The ESD under test is then excited bythe ETR, and a time record of the response is captured by the DAS.

Generalized Fast Summation Transformation (GFST).

Using Fast Summation Transformation (FST), it is possible to identifythe individual magnitude and phase for each of the constituents in asum-of-sines signal using an octave frequency step. With octaveharmonics, crosstalk error between the frequency components in the SOShas been completely eliminated. Consequently, the SOS waveform can berectified in two distinct ways (i.e., relative to the sine and then tothe cosine), thus generating two independent equations that are used tofind the magnitude and phase at each frequency of interest.

In a preferred embodiment of this invention, the Generalized FastSummation Transformation (GFST) algorithm is described, where thelogarithmic steps can be any integer, N, and not octaves (i.e., N=2).The use of harmonic frequency steps (i.e., integer multipliers) alwaysresults in perfect crosstalk rejection. As with FST, the capturedresponse time record must have a duration of at least one period of thelowest frequency, and the sampling frequency should be N^(k) timesfaster than the highest frequency in the SOS signal, where k>2 and aninteger. A flow diagram of the GFST implementation is provided in FIG. 1a and continued in FIG. 1 b.

Assuming a frequency step factor of r (i.e., N=r, where r>2), the GFSTanalysis consists of rectifying the response time record in fractions of1/r, resulting in r independent relationships. Since r can be greaterthan two, the system of equations will be over-specified and theimpedance calculation should be based on the pseudo-inverse (Mix, 1995)method. As with the FST algorithm, GFST can be used to identify theimpedance spectrum with a SOS input signal. The frequencies in this casewill consist of r harmonics. Note, however, that if r is a power of two(e.g., r=2²=4 or r=2³=8), then the FST algorithm can be applied instead.

The validity of this concept is based on the assumption that thesummation of the rectification of frequencies other than the one beingdetected goes to zero, and the pseudo-inverse exists. These assumptionswere verified mathematically and with software simulations and shown tobe true for SOS signals with a wide variation of amplitude and phaseshifts amongst the frequency components. Thus, GFST can be successfullyimplemented as a measure of ESD impedance using only one period of thelowest frequency.

Once the SOS signal with a frequency step factor of r has been appliedto the ESD, the general form of the captured time record is as given byEquation 1, where f_(OTHER)(n) refers to all of the frequency componentsexcept the one of interest.

$\begin{matrix}{{{TR}(n)} - {V_{P}{\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}} + {f_{other}(n)}} & (1)\end{matrix}$

Where:

-   -   V_(P) is the amplitude of the frequency of interest    -   TR is the time record    -   n is the discrete time step related to the sample time Δt

$N = \frac{T}{\Delta \; t}$

-   -   is the discrete period of the frequency of interest, where        -   Δt is the sample period        -   T is the time period of the frequency of interest    -   φ_(P)(n) is the phase angle of the frequency of interest    -   f_(other)(n) is all of the other frequency components in the        time record

To detect the desired frequency, Equation 1 is rectified as fractions ofthe period of that frequency, and the resulting time record is thensummed and averaged by the number of periods. The first rectificationplaces the first fraction of the period as positive and all the restnegative; the second rectification makes the second fraction positiveand all the rest negative; the r^(th) rectification makes the lastr^(th) fraction of the period as positive and all the rest negative. Forexample, Equation 2 shows that the summation term at the lowestfrequency (i.e., m₁) component has a positive rectification for thefirst 1/r of the period, and negative rectification for the rest.Equation 3 shows that summation term at the second lowest frequency(i.e., m₂) has a positive rectification for the second 1/r of theperiod, and negative rectification for the rest. This process isrepeated for all frequencies in the SOS signal. The summation term atthe highest frequency rectification (i.e., m_(r)) is described byEquation 4, where the last 1/r of the period is made positive and therest is negative.

$\begin{matrix}{m_{1} = {{V_{P}\begin{Bmatrix}{{\sum\limits_{n = 0}^{\frac{N}{r} - 1}\; {\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}} -} \\{\sum\limits_{n = {N/r}}^{N - 1}\; {\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}}\end{Bmatrix}} + \underset{= 0}{\underset{}{\sum\limits_{n = 0}^{\frac{N}{r} - 1}\; {f_{other}(n)}}} - \underset{= 0}{\underset{}{\sum\limits_{n = {N/r}}^{N - 1}\; {f_{other}(n)}}}}} & (2) \\{m_{2} = {{V_{P}\begin{Bmatrix}{{- {\sum\limits_{n = 0}^{\frac{N}{r} - 1}\; {\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}}} +} \\{{\sum\limits_{n = {N/r}}^{\frac{2N}{r} - 1}\; {\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}} -} \\{\sum\limits_{n = {2{N/r}}}^{N - 1}\; {\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}}\end{Bmatrix}} - \underset{= 0}{\underset{}{\sum\limits_{n = 0}^{\frac{N}{r} - 1}\; {f_{other}(n)}}} + \underset{= 0}{\underset{}{\sum\limits_{n = {N/r}}^{\frac{2N}{r} - 1}\; {f_{other}(n)}}} - \underset{= 0}{\underset{}{\sum\limits_{n = {2{N/r}}}^{N - 1}\; {f_{other}(n)}}}}} & (3) \\{m_{r} = {{V_{P}\begin{Bmatrix}{{- {\sum\limits_{n = 0}^{\frac{{({r - 1})}N}{r} - 1}\; {\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}}} +} \\{\sum\limits_{n = \frac{{({r - 1})}N}{r}}^{N - 1}\; {\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}}\end{Bmatrix}} - \underset{= 0}{\underset{}{\sum\limits_{n = 0}^{\frac{{({r - 1})}N}{r} - 1}\; {f_{other}(n)}}} + \underset{= 0}{\underset{}{\sum\limits_{n = \frac{{({r - 1})}N}{r}}^{N - 1}\; {f_{other}(n)}}}}} & (4)\end{matrix}$

Note that the summations of Equations 2, 3 and 4 are only over oneperiod of the frequency being detected. However, each frequencycomponent except the lowest frequency will contain multiple periods inthe overall time record and require correction. Let the number ofperiods in the time record for the frequency being detected be Q and thesum over the whole time record relative to Equations 2, 3 and 4 be S₁,S₂, and S_(r), respectively. Then, m₁, m₂, and m_(r) can be related toS₁, S₂, and S_(r) by Equation 5.

S₁=Qm_(i),S₂=Qm₂, . . . S_(r)=Qm_(r)  (5)

Using the identity of Equation 6, the summation terms m₁, m₂, and m_(r)can be modified as shown in Equations 7, 8, and 9.

$\begin{matrix}{{V_{P}{\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}} = {{V_{P}{\cos \left( \varphi_{P} \right)}{\sin \left( {\frac{2\pi}{N}n} \right)}} + {V_{P}{\sin \left( \varphi_{P} \right)}{\cos \left( {\frac{2\pi}{N}n} \right)}}}} & (6) \\{m_{1} = {V_{P}{\cos \left( \varphi_{P} \right)}\underset{K_{11}}{\underset{}{\begin{Bmatrix}{{\sum\limits_{n = 0}^{{N/r} - 1}\; {\sin \left( {\frac{2\pi}{N}n} \right)}} -} \\{\sum\limits_{n = {N/r}}^{N - 1}\; {\sin \left( {\frac{2\pi}{N}n} \right)}}\end{Bmatrix} +}}V_{P}{\sin \left( \varphi_{P} \right)}\underset{K_{12}}{\underset{}{\begin{Bmatrix}{{\sum\limits_{n = 0}^{{N/r} - 1}\; {\cos \left( {\frac{2\pi}{N}n} \right)}} -} \\{\sum\limits_{n = {N/r}}^{N - 1}\; {\cos \left( {\frac{2\pi}{N}n} \right)}}\end{Bmatrix}}}}} & (7) \\{m_{2} = {V_{P}{\cos \left( \varphi_{P} \right)}\underset{K_{21}}{\underset{}{\begin{Bmatrix}{{- {\sum\limits_{n = 0}^{\frac{N}{r} - 1}\; {\sin \left( {\frac{2\pi}{N}n} \right)}}} +} \\{{\sum\limits_{n = {N/r}}^{\frac{2N}{r} - 1}\; {\sin \left( {\frac{2\pi}{N}n} \right)}} -} \\{\sum\limits_{n = \frac{2N}{r}}^{N - 1}\; {\sin \left( {\frac{2\pi}{N}n} \right)}}\end{Bmatrix} +}}V_{P}{\sin \left( \varphi_{P} \right)}\underset{K_{22}}{\underset{}{\begin{Bmatrix}{{- {\sum\limits_{n = 0}^{\frac{N}{r} - 1}\; {\cos \left( {\frac{2\pi}{N}n} \right)}}} +} \\{{\sum\limits_{n = \frac{N}{r}}^{\frac{2N}{r} - 1}\; {\cos \left( {\frac{2\pi}{N}n} \right)}} -} \\{\sum\limits_{n = \frac{2N}{r}}^{N - 1}\; {\cos \left( {\frac{2\pi}{N}n} \right)}}\end{Bmatrix}}}}} & (8) \\{m_{r} = {V_{P}{\cos \left( \varphi_{P} \right)}\underset{K_{r\; 1}}{\underset{}{\begin{Bmatrix}{{- {\sum\limits_{n = 0}^{\frac{{({r - 1})}N}{r} - 1}\; {\sin \left( {\frac{2\pi}{N}n} \right)}}} +} \\{\sum\limits_{n = \frac{{({r - 1})}N}{r}}^{N - 1}\; {\sin \left( {\frac{2\pi}{N}n} \right)}}\end{Bmatrix} +}}V_{P}{\sin \left( \varphi_{P} \right)}\underset{K_{r\; 2}}{\underset{}{\begin{Bmatrix}{{\sum\limits_{n = 0}^{\frac{{({r - 1})}N}{r} - 1}\; {\cos \left( {\frac{2\pi}{N}n} \right)}} +} \\{\sum\limits_{n = \frac{{({r - 1})}N}{r}}^{N - 1}\; {\cos \left( {\frac{2\pi}{N}n} \right)}}\end{Bmatrix}}}}} & (9)\end{matrix}$

Equations 7, 8, and 9 can then be solved in matrix form as shown inEquation 10. The results can then be converted to polar form, as shownby Equation 11. This process is repeated for each frequency in the SOS.Using triads as an example (i.e., r=3), the matrix form of the GFSTimpedance calculation is shown in Equation 12.

$\begin{matrix}{\begin{bmatrix}{V_{P}\cos \; \varphi_{P}} \\{V_{P}\sin \; \varphi_{P}}\end{bmatrix} = {{\begin{Bmatrix}\begin{bmatrix}K_{11} & K_{21} & \ldots & K_{r\; 1} \\K_{12} & K_{22} & \ldots & K_{r\; 2}\end{bmatrix} \\\begin{bmatrix}K_{11} & K_{12} \\K_{21} & K_{22} \\\vdots & \vdots \\K_{r\; 1} & K_{r\; 2}\end{bmatrix}\end{Bmatrix}^{- 1}\begin{bmatrix}K_{11} & K_{21} & \ldots & K_{r\; 1} \\K_{12} & K_{22} & \ldots & K_{r\; 2}\end{bmatrix}}\begin{bmatrix}m_{1} \\m_{2} \\\vdots \\m_{r}\end{bmatrix}}} & (10) \\{\begin{matrix}{{V_{P}\sin \; \varphi_{P}} = C_{1}} \\{{V_{P}\cos \; \varphi_{P}} = C_{2}}\end{matrix},{{then}\text{:}\mspace{14mu} \begin{matrix}{V_{P} = \sqrt{C_{1}^{2} + C_{2}^{2}}} \\{\varphi_{P} = {\tan^{- 1}\left( \frac{C_{1}}{C_{2}} \right)}}\end{matrix}}} & (11) \\{\begin{bmatrix}{V_{P}\cos \; \varphi_{P}} \\{V_{P}\sin \; \varphi_{P}}\end{bmatrix} = {{\begin{Bmatrix}\begin{bmatrix}K_{1} & K_{3} & K_{5} \\K_{2} & K_{4} & K_{6}\end{bmatrix} \\\begin{bmatrix}K_{1} & K_{2} \\K_{3} & K_{4} \\K_{5} & K_{6}\end{bmatrix}\end{Bmatrix}^{- 1}\begin{bmatrix}K_{1} & K_{3} & K_{5} \\K_{2} & K_{4} & K_{6}\end{bmatrix}}\begin{bmatrix}m_{1} \\m_{2} \\m_{3}\end{bmatrix}}} & (12)\end{matrix}$

Reduced Time Fast Summation Transformation (RTFST).

It has been shown (Morrison, 2009) that it is possible to identify theindividual magnitude and phase of each of the constituents of a SOSsignal when the frequencies and the sample rate of the SOS are octaveharmonics using the Fast Summation Transformation method. When the FSTalgorithm is applied to the SOS for each frequency constituent, noisecrosstalk from the other frequencies present goes to zero regardless ofwhether those other frequencies are larger or smaller than the one beingdetected. Additionally, the captured time record length only has to beone period of the lowest frequency.

In one preferred embodiment of this invention, the time record for FSTcan be reduced to half a period of the lowest frequency using ReducedTime Fast Summation Transformation (RTFST). This can be accomplished bygenerating an octave harmonic SOS signal with all of the frequencycomponents except the second lowest frequency. After the signal isapplied to the ESD, the captured time record is also only a half periodof the lowest frequency, but it can be inverted and concatenated to theend of itself to generate a time record covering a full period. The FSTanalysis algorithm is then applied to detect the magnitude and phase forthe lowest frequency. To find the remaining frequency components, theresponse from the lowest frequency must first be changed back into ahalf-period time record and subtracted from the original captured timerecord. Since the remaining frequency components have more than oneperiod in the captured time record, the corresponding magnitudes andphases at each frequency can be detected with the FST algorithm. A flowdiagram of the RTFST implementation is provided in FIG. 2 a andcontinued in FIG. 2 b.

The resulting impedance spectrum will miss the second lowest frequency,but the measurement can be acquired in half the time as conventionalFST. If the missing frequency is desired, the process can be repeatedstarting with the second lowest frequency and skipping the third lowestfrequency. In other words, a new SOS excitation signal is generated,starting with the second lowest frequency and applied to the ESD. Thissecond measurement can be made in a quarter-period of the lowestfrequency. Thus, to generate a full impedance spectrum with all of thefrequency components, RTFST can be used to complete the measurement morequickly than FST since it requires only a total of three-quarters of aperiod of the lowest frequency (i.e., half a period for the initialmeasurement plus a quarter-period to find the second lowest frequencycomponent).

It should be noted that any DC bias in the measurement other than thehalf-period of the lowest frequency will generate errors in theanalysis. For example, the concatenation process will transform thatcorrupting DC into error at the lowest frequency and yield inaccurateresults. The hardware used to implement this technique must, thereforefilter out any DC bias prior to capturing the ESD response to an RTFSTSOS signal that is applied.

The derivation of RTFST starts with the digitized acquired time recordas a discrete function, TR(n). The time record is a SOS signal that hasa duration of a half-period of the lowest frequency with unknownmagnitudes and phases for each of the constituents. The SOS isconstrained to be compatible with the FST algorithm (Morrison, 2009),except that the second frequency is omitted. Recall for FST, the SOSfrequencies go as octave harmonics and the sample frequency is alsooctave harmonic (e.g., at least 4 times faster than the highestfrequency in the SOS). The following are defined:

$\begin{matrix}{{T = {N\; \Delta \; t}}{{{TR}\left( {n\; \Delta \; t} \right)}\overset{\Delta}{=}{{TR}(n)}}{{P(n)}\overset{\Delta}{=}{{u(n)} - {u\left( {n - \frac{N}{2}} \right)}}}{{P\left( {n - \frac{N}{2}} \right)}\overset{\Delta}{=}{{u\left( {n - \frac{N}{2}} \right)} - {u\left( {n - N} \right)}}}{{{PP}(n)}\overset{\Delta}{=}{{u(n)} - {u\left( {n - N} \right)}}}} & (13)\end{matrix}$

Where:

-   -   Δt is the sample time    -   T is the period of the lowest frequency in the SOS    -   n is discrete time    -   N is the discrete period of the lowest frequency in the SOS    -   u(n) is the discrete time step

TR(n) is described in Equation 14 as the product of discrete sine wavesand pulse functions, P(n), where f_(OTHER)(n) refers to all of thefrequency components except the one of interest. The pulse functiontruncates the sine wave at the lowest frequency to a half-period asshown in Equation 13 above. Observe that it is assumed there is no DCpresent in the time record other than from the half period of the firstfrequency.

$\begin{matrix}{{{TR}(n)} = {\left\lbrack {{V_{P}{\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}} + {f_{OTHER}(n)}} \right\rbrack {P(n)}}} & (14)\end{matrix}$

Where: V_(P), φ_(P) (i.e., magnitude and phase) need to be detected.

Equation 15 gives an expression for the adjusted time record, where theacquired time record TR(n) for the lowest frequency component isinverted and concatenated to the end of itself. Equation 15 can then besimplified into Equation 16, with the f_(OTHER)(n) components. The FSTalgorithm (Morrison, 2009) is then applied to Equation 16 by firstrectifying relative to the sine and summing over the whole recordlength, as shown in Equation 17.

$\begin{matrix}{{{TR}(n)} = {{\begin{bmatrix}{{V_{P}{\sin \begin{pmatrix}{{\frac{2\pi}{N}n} +} \\\varphi_{P}\end{pmatrix}}} +} \\{f_{OTHER}(n)}\end{bmatrix}{P(n)}} - {\begin{bmatrix}{{V_{P}{\sin \begin{pmatrix}{{\frac{2\pi}{N}\left( {n - \frac{N}{2}} \right)} +} \\\varphi_{P}\end{pmatrix}}} +} \\{f_{OTHER}\left( {n - \frac{N}{2}} \right)}\end{bmatrix}{P\left( {n - \frac{N}{2}} \right)}}}} & (15) \\{{{TR}(n)} = {{\left\lbrack {V_{P}{\sin \begin{pmatrix}{{\frac{2\pi}{N}n} +} \\\varphi_{P}\end{pmatrix}}} \right\rbrack {{PP}(n)}} + {{f_{OTHER}(n)}{P(n)}} - {{f_{OTHER}\left( {n - \frac{N}{2}} \right)}{P\left( {n - \frac{N}{2}} \right)}}}} & (16) \\{{m_{1} = {{\sum\limits_{n = 0}^{\frac{N}{2} - 1}\; {V_{P}{\sin \begin{pmatrix}{{\frac{2\pi}{N}n} +} \\\varphi_{P}\end{pmatrix}}}} - {\sum\limits_{n = \frac{N}{2}}^{N - 1}\; {V_{P}{\sin \begin{pmatrix}{{\frac{2\pi}{N}n} +} \\\varphi_{P}\end{pmatrix}}}} + \underset{\underset{= 0}{}}{\sum\limits_{n = 0}^{\frac{N}{2} - 1}{f_{OTHER}(n)}} - \underset{\underset{= 0}{}}{\sum\limits_{n = \frac{N}{2}}^{N - 1}{f_{OTHER}\left( {n - \frac{N}{2}} \right)}}}},} & (17) \\{\mspace{79mu} {m_{1} = {{\sum\limits_{n = 0}^{\frac{N}{2} - 1}\; {V_{P}{\sin \begin{pmatrix}{{\frac{2\pi}{N}n} +} \\\varphi_{P}\end{pmatrix}}}} - {\sum\limits_{n = \frac{N}{2}}^{N - 1}\; {V_{P}{\sin \begin{pmatrix}{{\frac{2\pi}{N}n} +} \\\varphi_{P}\end{pmatrix}}}}}}} & \;\end{matrix}$

The summation of f_(OTHER) in Equation 17 will be zero as it is over aninteger number of periods of each frequency within f_(OTHER). Since thenext frequency in the SOS signal is constrained to be at least fourtimes the first (i.e., the second lowest frequency was omitted), thesmallest frequency present in f_(OTHER) will have at least two fullperiods over the sum, and thus that sum will always be zero. Thesummation for the cosine rectification is given by Equation 18. As withthe sine rectification, the summation of f_(OTHER) in Equation 18 willalso always be zero since it includes multiples of a full period at eachfrequency component within the half-period duration of the lowestfrequency.

$\begin{matrix}{{m_{2} = {{\sum\limits_{n = 0}^{\frac{N}{4} - 1}{V_{P}{\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}}} - {\sum\limits_{n = \frac{N}{4}}^{\frac{3N}{4} - 1}{V_{P}{\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}}} + {\sum\limits_{n = \frac{3N}{4}}^{N - 1}{V_{P}{\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}}} + \underset{\underset{= 0}{}}{\sum\limits_{n = 0}^{\frac{N}{4} - 1}{f_{OTHER}(n)}} - \underset{\underset{= 0}{}}{\sum\limits_{n = \frac{N}{4}}^{\frac{N}{2} - 1}{f_{OTHER}\left( {n - \frac{N}{2}} \right)}} + \underset{\underset{= 0}{}}{\sum\limits_{n = \frac{N}{2}}^{\frac{3N}{4} - 1}{f_{OTHER}(n)}} - \underset{\underset{= 0}{}}{\sum\limits_{n = \frac{3N}{4}}^{N - 1}{f_{OTHER}\left( {n - \frac{N}{2}} \right)}}}},{m_{2} = {{\sum\limits_{n = 0}^{\frac{N}{4} - 1}{V_{P}{\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}}} - {\sum\limits_{n = \frac{N}{4}}^{\frac{3N}{4} - 1}{V_{P}\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}} + {\sum\limits_{n = \frac{3N}{4}}^{N - 1}{V_{P}{\sin \left( {{\frac{2\pi}{N}n} + \varphi_{P}} \right)}}}}}} & (18)\end{matrix}$

The trig identity of Equation 6 is applied to Equations 17 and 18, andthe resulting summations are shown in Equations 19 and 20. Theseequations are subsequently solved for V_(P), φ_(P) using the matrixcalculation for the FST algorithm, as shown by Equations 21 through 23.

$\begin{matrix}{m_{1} = {{V_{P}{\cos \left( \varphi_{P} \right)}\underset{\underset{K_{1}}{}}{\left\{ {{\sum\limits_{n = 0}^{\frac{N}{2} - 1}{\sin \left( {\frac{2\pi}{N}n} \right)}} - {\sum\limits_{n = \frac{N}{2}}^{N - 1}{\sin \left( {\frac{2\pi}{N}n} \right)}}} \right\}}} + {V_{P}{\sin \left( \varphi_{P} \right)}\underset{\underset{K_{2}}{}}{\left\{ {{\sum\limits_{n = 0}^{\frac{N}{2} - 1}{\cos \left( {\frac{2\pi}{N}n} \right)}} - {\sum\limits_{n = \frac{N}{2}}^{N - 1}{\cos \left( {\frac{2\pi}{N}n} \right)}}} \right\}}}}} & (19) \\{m_{2} = {{V_{P}{\cos \left( \varphi_{P} \right)}\underset{\underset{K_{3}}{}}{\left\{ {{\sum\limits_{n = 0}^{\frac{N}{4} - 1}{\sin \left( {\frac{2\pi}{N}n} \right)}} - {\sum\limits_{n = \frac{N}{4}}^{\frac{3N}{4} - 1}{\sin \left( {\frac{2\pi}{N}n} \right)}} + {\sum\limits_{n = \frac{3N}{4}}^{N - 1}{\sin \left( {\frac{2\pi}{N}n} \right)}}} \right\}}} + {V_{P}{\sin \left( \varphi_{P} \right)}\underset{\underset{K_{4}}{}}{\left\{ {{\sum\limits_{n = 0}^{\frac{N}{3} - 1}{\cos \left( {\frac{2\pi}{N}n} \right)}} - {\sum\limits_{n = \frac{N}{4}}^{\frac{3N}{4} - 1}{\cos \left( {\frac{2\pi}{N}n} \right)}} + {\sum\limits_{n = \frac{3N}{4}}^{N - 1}{\cos \left( {\frac{2\pi}{N}n} \right)}}} \right\}}}}} & (20) \\\left. \begin{matrix}{m_{1} = {{V_{P}\cos \; \varphi \; K_{1}} + {V_{P}\sin \; \varphi \; K_{2}}}} \\{m_{2} = {{V_{P}\cos \; \varphi \; K_{3}} + {V_{P}\sin \; \varphi \; K_{4}}}}\end{matrix}\Rightarrow{\begin{bmatrix}m_{1} \\m_{2}\end{bmatrix}{{\begin{matrix} = \\ = \end{matrix}\begin{bmatrix}{K_{1},K_{2}} \\{K_{3},K_{4}}\end{bmatrix}}\begin{bmatrix}{V_{P}\cos \; \varphi} \\{V_{P}\sin \; \varphi}\end{bmatrix}}} \right. & (21) \\{\begin{bmatrix}{V_{P}\cos \; \varphi} \\{V_{P}\sin \; \varphi}\end{bmatrix}{{\begin{matrix} = \\ = \end{matrix}\begin{bmatrix}{\frac{K_{4}}{{K_{1}K_{4}} - {K_{2}K_{3}}},\frac{K_{2}}{{K_{2}K_{3}} - {K_{1}K_{4}}}} \\{\frac{K_{3}}{{K_{2}K_{3}} - {K_{1}K_{4}}},\frac{K_{1}}{{K_{1}K_{4}} - {K_{2}K_{3}}}}\end{bmatrix}}\begin{bmatrix}m_{1} \\m_{2}\end{bmatrix}}} & (22) \\{{{Let}\text{:}\mspace{14mu} \begin{matrix}{{V_{P}\sin \; \varphi} = C_{1}} \\{{{V_{P}\cos \; \varphi} = C_{2}},}\end{matrix}}{{then}\text{:}\mspace{14mu} \begin{matrix}{V_{P} = \sqrt{C_{1}^{2} + C_{2}^{2}}} \\{\varphi = {\tan^{- 1}\left( \frac{C_{1}}{C_{2}} \right)}}\end{matrix}}} & (23)\end{matrix}$

This RTFST development first identifies the magnitude and phase, V_(P),φ_(P) of the lowest frequency component of the SOS signal (TR(n)) with aduration of a half-period. Because that half-period is not orthogonal toany of the higher frequencies, it must be subtracted from TR(n) beforethe magnitude and phase for the remaining frequency components can beidentified. The detected magnitude and phase (V_(P), φ_(P)) from thelowest frequency component are used to build a time record that is ahalf-period in length and it is subtracted from the original TR(n). Thestandard FST algorithm can then be used to process and identify all theremaining frequency components. The RTFST was tested with softwaresimulations and shown to be true for SOS signals with a wide variationof amplitude and phase shifts amongst the frequency components. Thus,RTFST can also be successfully implemented as a measure of ESD impedanceusing only half the period of the lowest frequency.

The FST algorithm has been previously described (Morrison, 2009) and issummarized as follows:

Equation 24 (same as Equation 6) represents the sampled signal componentat a specific frequency that is to be detected. The amplitude, V_(P) andphase, φ at each frequency component are the desired information.

$\begin{matrix}{{V_{P}{\sin \left( {{\frac{2\pi}{N}n} + \varphi} \right)}} = {{V_{P}{\sin \left( {\frac{2\pi}{N}n} \right)}\cos \; \varphi} + {V_{P}{\cos \left( {\frac{2\pi}{N}n} \right)}\sin \; \varphi}}} & (24)\end{matrix}$

Where:

V_(P) is amplitude response of the frequency of interest.

N is the number of samples over a period of the frequency of interest.

φ is the phase response of the frequency of interest.

n is the discrete time index.

In Equation 24, N must be constrained as log₂ (N), and must be aninteger greater than one. Additionally, the frequency of interest isgiven as:

$\begin{matrix}{f = \frac{1}{N\; \Delta \; t}} & (25)\end{matrix}$

Where: Δt is the sample period

In Equations 26 through 28, the signal has been rectified relative to asine wave of the frequency of interest, and all the sample values aresummed together. In Equation 29 through 31, the signal has beenrectified relative to the cosine wave of the frequency of interest, andagain the samples are summed. Observe that rectification simply involveschanging the sign of the sample values relative to the sine wave orcosine wave timing.

$\begin{matrix}{m_{1} = {{V_{P}{\sum\limits_{n = 0}^{\frac{N}{2} - 1}{\sin \left( {{\frac{2\pi}{N}n} + \varphi} \right)}}} - {V_{P}{\sum\limits_{n = \frac{N}{2}}^{N - 1}{\sin \left( {{\frac{2\pi}{N}n} + \varphi} \right)}}}}} & (26) \\{m_{1} = {V_{P}{\sum\limits_{n = 0}^{\frac{N}{2} - 1}\begin{matrix}{\left( {{{\sin \left( {\frac{2\pi}{N}n} \right)}\cos \; \varphi} + {{\cos \left( {\frac{2\pi}{N}n} \right)}\sin \; \varphi}} \right) -} \\{V_{P}{\sum\limits_{n = \frac{N}{2}}^{N - 1}\left( {{{\sin \left( {\frac{2\pi}{N}n} \right)}\cos \; \varphi} + {{\cos \left( {\frac{2\pi}{N}n} \right)}\sin \; \varphi}} \right)}}\end{matrix}}}} & (27) \\{m_{1} = {{V_{P}\cos \; \varphi \underset{\underset{K_{1}}{}}{\left\lbrack {{\sum\limits_{n = 0}^{\frac{N}{2} - 1}\left( {\sin \left( {\frac{2\pi}{N}n} \right)} \right)} - {\sum\limits_{n = \frac{N}{2}}^{N - 1}\left( {\sin \left( {\frac{2\pi}{N}n} \right)} \right)}} \right\rbrack}} + {V_{P}\sin \; \varphi \underset{\underset{K_{2}}{}}{\left\lbrack {{\sum\limits_{n = 0}^{\frac{N}{2} - 1}\left( {\cos \left( {\frac{2\pi}{N}n} \right)} \right)} - {\sum\limits_{n = \frac{N}{2}}^{N - 1}\left( {\cos \left( {\frac{2\pi}{N}n} \right)} \right)}} \right\rbrack}}}} & (28) \\{m_{2} = {{V_{P}{\sum\limits_{n = 0}^{\frac{N}{4} - 1}{\sin \left( {{\frac{2\pi}{N}n} + \varphi} \right)}}} - {V_{P}{\sum\limits_{n = \frac{N}{4}}^{{3\frac{N}{4}} - 1}{\sin \left( {{\frac{2\pi}{N}n} + \varphi} \right)}}} + {V_{P}{\sum\limits_{n = {3\frac{N}{4}}}^{N - 1}{\sin \left( {{\frac{2\pi}{N}n} + \varphi} \right)}}}}} & (29) \\{m_{2} = {{V_{P}{\sum\limits_{n = 0}^{\frac{N}{4} - 1}\left( {{{\sin \left( {\frac{2\pi}{N}n} \right)}\cos \; \varphi} + {{\cos \left( {\frac{2\pi}{N}n} \right)}\sin \; \varphi}} \right)}} - {V_{P}{\sum\limits_{n = \frac{N}{4}}^{{3\frac{N}{4}} - 1}\left( {{{\sin \left( {\frac{2\pi}{N}n} \right)}\cos \; \varphi} + {{\cos \left( {\frac{2\pi}{N}n} \right)}\sin \; \varphi}} \right)}} + {V_{P}{\sum\limits_{n = {3\frac{N}{4}}}^{N - 1}\left( {{{\sin \left( {\frac{2\pi}{N}n} \right)}\cos \; \varphi} + {{\cos \left( {\frac{2\pi}{N}n} \right)}\sin \; \varphi}} \right)}}}} & (30) \\{m_{2} = {{V_{P}\cos \; \varphi \underset{\underset{K_{3}}{}}{\left\lbrack {{\sum\limits_{n = 0}^{\frac{N}{4} - 1}\left( {\sin \left( {\frac{2\pi}{N}n} \right)} \right)} - {\sum\limits_{n = \frac{N}{4}}^{{3\frac{N}{4}} - 1}\left( {\sin \left( {\frac{2\pi}{N}n} \right)} \right)} + {\sum\limits_{n = {3\frac{N}{4}}}^{N - 1}\left( {\sin \left( {\frac{2\pi}{N}n} \right)} \right)}} \right\rbrack}} + {V_{P}\sin \; \varphi \underset{\underset{K_{4}}{}}{\left\lbrack {{\sum\limits_{n = 0}^{\frac{N}{4} - 1}\left( {\cos \left( {\frac{2\pi}{N}n} \right)} \right)} - {\sum\limits_{n = \frac{N}{4}}^{{3\frac{N}{4}} - 1}\left( {\cos \left( {\frac{2\pi}{N}n} \right)} \right)} + {\sum\limits_{n = {3\frac{N}{4}}}^{N - 1}\left( {\cos \left( {\frac{2\pi}{N}n} \right)} \right)}} \right\rbrack}}}} & (31)\end{matrix}$

Note that the parameters K₁, K₂, K₃, K₄ are known for each frequency andm₁, m₂ are the numerical result of the rectifying algorithm for eachfrequency. The magnitude and phase at each frequency can then beobtained as shown in Equations 32 through 34.

$\begin{matrix}\left. \begin{matrix}{m_{1} = {{V_{P}\cos \; \varphi \; K_{1}} + {V_{P}\sin \; \varphi \; K_{2}}}} \\{m_{2} = {{V_{P}\cos \; \varphi \; K_{3}} + {V_{P}\sin \; \varphi \; K_{4}}}}\end{matrix}\Rightarrow{\begin{bmatrix}m_{1} \\m_{2}\end{bmatrix}{{\begin{matrix} = \\ = \end{matrix}\begin{bmatrix}{K_{1},K_{2}} \\{K_{3},K_{4}}\end{bmatrix}}\begin{bmatrix}{V_{P}\cos \; \varphi} \\{V_{P}\sin \; \varphi}\end{bmatrix}}} \right. & (32) \\{\begin{bmatrix}{V_{P}\cos \; \varphi} \\{V_{P}\sin \; \varphi}\end{bmatrix}{{\begin{matrix} = \\ = \end{matrix}\begin{bmatrix}{\frac{K_{4}}{{K_{1}K_{4}} - {K_{2}K_{3}}},\frac{K_{2}}{{K_{2}K_{3}} - {K_{1}K_{4}}}} \\{\frac{K_{3}}{{K_{2}K_{3}} - {K_{1}K_{4}}},\frac{K_{1}}{{K_{1}K_{4}} - {K_{2}K_{3}}}}\end{bmatrix}}\begin{bmatrix}m_{1} \\m_{2}\end{bmatrix}}} & (33) \\{{{Let}\text{:}\mspace{14mu} \begin{matrix}{{V_{P}\sin \; \varphi} = C_{1}} \\{{{V_{P}\cos \; \varphi} = C_{2}},}\end{matrix}}{{then}\text{:}\mspace{14mu} \begin{matrix}{V_{P} = \sqrt{C_{1}^{2} + C_{2}^{2}}} \\{\varphi = {\tan^{- 1}\left( \frac{C_{1}}{C_{2}} \right)}}\end{matrix}}} & (34)\end{matrix}$

When detecting the magnitude and phase at a given frequency of interest,all of the other frequency components sum to zero. Thus, for thistechnique, the crosstalk noise is eliminated, and the SOS time recordlength can be as short as one period of the lowest frequency. However,the FST method only works if the octave harmonic relationship holds forall frequencies in the SOS (including the sample frequency). Thisensures that there will always be an even number of samples over aperiod of any frequency present in the SOS. For rectification, the timerecord is multiplied by the rectification function relative to the sineand cosine (i.e., change the signs). This process is repeated to obtainthe magnitude and phase for each selected frequency to obtain theoverall impedance spectrum for the ESD. However, for frequencies higherthan the first frequency there will be multiple (Q) periods of thatfrequency and the resulting summations for m₁ and m₂ must be normalizedto Q

It is understood that the foregoing examples are merely illustrative ofthe present invention. Certain modifications of the articles and/ormethods employed may be made and still achieve the objectives of theinvention. Such modifications are contemplated as within the scope ofthe claimed invention.

REFERENCES

-   Christophersen, J. P., D. F. Glenn, C. G. Motloch, R. B.    Wright, C. D. Ho, and V. S. Battaglia, “Electrochemical Impedance    Spectroscopy Testing on the Advanced Technology Development Program    Lithium-Ion Cells,” IEEE Trans. Veh. Technol., 56(3), 1851-1855    (2002).-   Morrison, J. M., B. Smyth, J. Wold, D. K. Butherus, W. H.    Morrison, J. P. Christophersen, C. G. Motloch, “Fast Summation    Transformation for Battery Impedance Identification.” Proceedings    from the IEEE Aerospace Conference (2009).-   Mix, Dwight F., “Random Signal Processing”, Page 296; Prentice Hall    Publishing Company, (1995).-   Christophersen, J. P., et al. “Impedance Noise Identification for    State-of-Health Prognostics,” 43rd Power Sources Conference, July    7-10, Philadelphia, Pa. (2008).-   Christophersen, J. P., C. G. Motloch, C. D. Ho, J. L.    Morrison, R. C. Fenton, V. S. Battaglia, and T. Q. Duong, “Lumped    Parameter Modeling as a Predictive Tool for a Battery Status    Monitor.” Proceedings from 2003 IEEE Vehicular Technology    Conference, October (2003).-   Morrison, J. L. and W. H. Morrison, “Real Time Estimation of Battery    Impedance,” Proceedings from the IEEE Aerospace Conference (2006).

Chapra, Steven C., and Raymond P. Canale. “Numerical Methods ForEngineers” Pages 394-8, McGraw-Hill Publishing Company, (1985)

-   Smyth, Brian, “Development of a Real Time Battery Impedance    Measuring System” M.S. Thesis Montana Tech of the University of    Montana; (2008)

1. A method for detecting system function of an energy storage deviceunder test by measuring frequency response, the method comprising thesteps of: (a) generating a vector of frequencies, by a method comprisingthe steps of (1) selecting a lowest frequency, (2) selecting an integerstep factor, (3) multiplying the lowest frequency by the integer stepfactor to obtain a second frequency, (4) multiplying the secondfrequency by the integer step factor to obtain a next frequency, and (5)repeating step (4) until each frequency in the vector of frequencies isgenerated; (b) generating a vector of amplitudes and phases; (c)assembling a wideband excitation signal using the vector of frequenciesand the vector of amplitudes and phases; (d) conditioning the excitationsignal to be compatible with the energy storage device under test; (e)exciting the energy storage device under test with the conditionedexcitation signal for one period of the lowest frequency; (f) capturinga response time record with a data acquisition system; and (g)processing the response time record to obtain the frequency response. 2.The method of claim 1, wherein said integer step factor is greater than2.
 3. The method of claim 1, wherein said excitation signal is a sum ofsines excitation signal.
 4. The method of claim 3, wherein said responsetime record is captured at a sampling rate that is step factor harmonicto said sum of sines excitation signal and is at least said step factorsquared times a highest frequency in said sum of sines excitationsignal.
 5. The method of claim 1, wherein said response time record isprocessed by rectifying said response time record relative to variousforms.
 6. The method of claim 5, wherein the response time record isrectified with r forms, where r is said integer step factor, and N isthe number of samples over the period (i.e., n=n₁, n₂, . . . , N), thefirst form of the rectification is: ${R_{1}(n)} = \begin{bmatrix}{1,} & {0 \leq n < \frac{N}{r}} \\{{- 1},} & {\frac{N}{r} \leq n < N}\end{bmatrix}$ For the i^(th) ' rectification, where 1<i<r, the form isas follows: ${R_{i}(n)} = \begin{bmatrix}{{- 1},} & {0 \leq n < \frac{\left( {i - 1} \right)N}{r}} \\{1,} & {\frac{\left( {i - 1} \right)N}{r} \leq n < \frac{iN}{r}} \\{{- 1},} & {\frac{iN}{r} \leq n < N}\end{bmatrix}$ Finally, for the r^(th) rectification, the form is asfollows: ${R_{r}(n)} = \begin{bmatrix}{{- 1},} & {0 \leq n < \frac{\left( {r - 1} \right)N}{r}} \\{1,} & {\frac{\left( {r - 1} \right)N}{r} \leq n < N}\end{bmatrix}$ to obtain a rectified signal.
 7. The method of claim 6,wherein each of said r forms are summed up to obtain r different sumsincluding m₁ . . . m_(r).
 8. The method of claim 7, wherein saidsummation terms (m₁ . . . m_(r)) are normalized to said response timerecordS₁=Qm₁,S₂=Qm₂S_(r)=Qm_(r) Where m₁ . . . m_(r) are summation terms ofsaid rectified signal Q is the number of periods in said response timerecord for a frequency being detected S₁, S₂, . . . , S_(r) arecorrected summation terms.
 9. The method of claim 8, wherein saidrectified signal is used to calculate magnitude and phase of saidresponse time record with $\begin{bmatrix}{V_{P}\cos \; \varphi_{P}} \\{V_{P}\sin \; \varphi_{P}}\end{bmatrix} = {{\left\{ {\begin{bmatrix}K_{11} & K_{21} & \cdots & K_{r\; 1} \\K_{12} & K_{22} & \cdots & K_{r\; 2}\end{bmatrix}\begin{bmatrix}{K_{11}K_{12}} \\{K_{21}K_{22}} \\{\vdots\vdots} \\{K_{r\; 1}K_{r\; 2}}\end{bmatrix}} \right\}^{- 1}\begin{bmatrix}K_{11} & K_{21} & \cdots & K_{r\; 1} \\K_{12} & K_{22} & \cdots & K_{r\; 2}\end{bmatrix}}\begin{bmatrix}m_{1} \\m_{2} \\\vdots \\m_{r}\end{bmatrix}}$ Where V_(P) is a desired magnitude φ_(P) is a desiredphase m₁ . . . m_(r) are summation terms of the rectified signal K₁₁ . .. K_(r1) are summations of the rectified sine term K₁₂ . . . K_(r2) aresummations of the rectified cosine term.
 10. The method of claim 9,wherein said magnitude and phase of said response time record arecalculated at each frequency interesting said vector of frequencies togenerate an impedance spectrum of said frequency response.
 11. A methodfor detecting system function of an energy storage device under test bymeasuring frequency response, the method comprising the steps of: (a)generating a vector of frequencies, by a method comprising the steps of(1) selecting a lowest frequency, (2) selecting an integer step factor,(3) multiplying the lowest frequency by the integer step factor toobtain a second frequency, (4) multiplying the second frequency by theinteger step factor to obtain a third frequency, (5) multiplying thethird frequency by the integer step factor to obtain a next frequency,(6) repeating step (5) until each frequency in the vector of frequenciesis generated, and (7) removing the second frequency from the vector offrequencies; (b) generating a vector of amplitudes and phases; (c)assembling a wideband excitation signal using the vector of frequenciesand the vector of amplitudes and phases; (d) conditioning the excitationsignal to be compatible with the energy storage device under test; (e)exciting the energy storage device under test with the conditionedexcitation signal for one-half period of the lowest frequency; (f)capturing a response time record with a data acquisition system; and (g)processing the response time record to obtain the frequency response.12. The method of claim 11, wherein said integer step factor is a factorof two (i.e., r=2, 4, 8, 16, etc.).
 13. The method of claim 11, whereinsaid excitation signal is a sum of sines excitation signal.
 14. Themethod of claim 13, wherein said response time record is captured at asampling rate that is octave harmonic to said sum of sines excitationsignal and is at least four times a highest frequency in the sum ofsines excitation signal.
 15. The method of claim 11, wherein thecaptured response time record is inverted and concatenated to saidcaptured response time record.
 16. The method of claim 11, whereinresponse time record is processed by rectifying the response time recordrelative to sine and cosine.
 17. The method of claim 16, wherein saidresponse time record is rectified using a method selected from the groupconsisting of non-zero crossing, with N as the number of samples overthe period the sine wave form of the rectification is:${{Rs}(n)} = \begin{bmatrix}{1,} & {0 \leq n < \frac{N}{2}} \\{{- 1},} & {\frac{N}{2} \leq n < N}\end{bmatrix}$ The cosine form is: ${{Rc}(n)} = \begin{bmatrix}{1,} & {0 \leq n < \frac{N}{4}} \\{{- 1},} & {\frac{N}{4} \leq n < \frac{3N}{4}} \\{1,} & {\frac{3N}{4} \leq n < N}\end{bmatrix}$ and, zero crossing, with N as the number of samples overthe period the sine wave form of the rectification is:${{Rs}(n)} = \begin{bmatrix}{0,} & {n = 0} \\{1,} & {0 < n < \frac{N}{2}} \\{0,} & {n = \frac{n}{2}} \\{{- 1},} & {\frac{N}{2} < n < N}\end{bmatrix}$ The cosine form is: ${{Rs}(n)} = \begin{bmatrix}{1,} & {0 \leq n < \frac{N}{4}} \\{0,} & {n = \frac{N}{4}} \\{{- 1},} & {\frac{N}{4} < n < \frac{3N}{4}} \\{0,} & {n = \frac{3N}{4}} \\{1,} & {\frac{3N}{4} < n < N}\end{bmatrix}$ to obtain a rectified signal.
 18. The method of claim 17,wherein said rectified signal is used to calculate magnitude and phaseof said response time record with $\begin{bmatrix}{V_{P}\cos \; \varphi} \\{V_{P}\sin \; \varphi}\end{bmatrix}{{\begin{matrix} = \\ = \end{matrix}\begin{bmatrix}{\frac{K_{4}}{{K_{1}K_{4}} - {K_{2}K_{3}}},\frac{K_{2}}{{K_{2}K_{3}} - {K_{1}K_{4}}}} \\{\frac{K_{3}}{{K_{2}K_{3}} - {K_{1}K_{4}}},\frac{K_{1}}{{K_{1}K_{4}} - {K_{2}K_{3}}}}\end{bmatrix}}\begin{bmatrix}m_{1} \\m_{2}\end{bmatrix}}$ Where V_(P) is the desired magnitude φ_(P) is thedesired phase m₁, m₂ are summation terms of the rectified signal K₁, K₂are summations of the rectified sine term K₃, K₄ are summations of therectified cosine term.
 19. The method of claim 18, wherein saidcalculated magnitude and phase of said lowest frequency are used togenerate a half-period time record that is subtracted from said responsetime record.
 20. The method of claim 17, wherein said summation terms(m₁ . . . m_(r)) are normalized to said response time record.S₁=Qm₁,S₂=Qm₂, . . . S_(r)=Qm, Where m₁ . . . m_(r) are summation termsof the rectified signal Q is the number of periods in the time recordfor the frequency being detected S₁, S₂, . . . S_(r) are the correctedsummation terms.
 21. The method of claim 20, wherein the magnitude andphase response are calculated at each frequency in said vector offrequencies to generate an impedance spectrum of said frequencyresponse.
 22. The method of claim 10, further comprising the step of:(h) repeating steps (a)-(g), except wherein in steps (a)(7) said lowestand said third frequency are removed from the vector of frequencies, andin step (e) the energy storage device under test is excited forone-quarter period of said lowest frequency; and combining saidfrequency response from step (g) and the frequency response from step(h) to obtain a frequency response with a duration of three-quarters ofa period of said lowest frequency.